Sobolev-Trace inequalities of order four

TL;DR

This paper establishes sharp Sobolev inequalities of order four on Euclidean d-balls for d ≥ 4, generalizing classical inequalities and utilizing scattering theory on hyperbolic balls, with applications to extremals in log-determinant formulas.

Contribution

It introduces new sharp Sobolev inequalities of order four on Euclidean balls using scattering theory, extending classical results and characterizing extremals in conformal geometry.

Findings

Established sharp Sobolev inequalities of order four for d ≥ 4.

Generalized the Lebedev-Milin inequality for d=4.

Characterized extremals of the log-determinant formula.

Abstract

We establish sharp Sobolev inequalities of order four on Euclidean d-balls for d greater than or equal to four. When d=4, our inequality generalizes the classical second order Lebedev-Milin inequality on Euclidean 2-balls. Our method relies on the use of scattering theory on hyperbolic d-balls. As an application, we charcaterize the extremals of the main term in the log-determinant formula corresponding to the conformal Laplacian coupled with the boundary Robin operator on Euclidean 4-balls.